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Markov random fields, Gibbs states and entropy minimality Chandgotia, Nishant


The well-known Hammersley-Clifford Theorem states (under certain conditions) that any Markov random field is a Gibbs state for a nearest neighbour interaction. Following Petersen and Schmidt we utilise the formalism of cocycles for the homoclinic relation and introduce "Markov cocycles", reparametrisations of Markov specifications. We exploit this formalism to deduce the conclusion of the Hammersley-Clifford Theorem for a family of Markov random fields which are outside the theorem's purview (including Markov random fields whose support is the d-dimensional "3-colored chessboard"). On the other extreme, we construct a family of shift-invariant Markov random fields which are not given by any finite range shift-invariant interaction. The techniques that we use for this problem are further expanded upon to obtain the following results: Given a "four-cycle free" finite undirected graph H without self-loops, consider the corresponding 'vertex' shift, H ơm(Zd,H) denoted by X(H). We prove that X(H) has the pivot property, meaning that for all distinct configurations x,y ∈ X(H) which differ only at finitely many sites there is a sequence of configurations (x=x¹),x²,...,(xn =y) ∈ X(H) for which the successive configurations (xi,xi+1) differ exactly at a single site. Further if H is connected we prove that X(H) is entropy minimal, meaning that every shift space strictly contained in X(H) has strictly smaller entropy. The proofs of these seemingly disparate statements are related by the use of the 'lifts' of the configurations in X(H) to their universal cover and the introduction of 'height functions' in this context. Further we generalise the Hammersley-Clifford theorem with an added condition that the underlying graph is bipartite. Taking inspiration from Brightwell and Winkler we introduce a notion of folding for configuration spaces called strong config-folding to prove that if all Markov random fields supported on X are Gibbs with some nearest neighbour interaction so are Markov random fields supported on the "strong config-folds" and "strong config-unfolds" of X.

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